In this paper, the separation transformation approach is extended to the （N ＋ 1）-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of SHe s...In this paper, the separation transformation approach is extended to the （N ＋ 1）-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of SHe superfluid. This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation. Then the general solutions of the set of partial differential equations are obta/ned and the nonlinear ordinary differential equation is solved by F-expansion method. Finally, many new exact solutions of the （N ＋ 1）-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation. For the case of N 〉 2, there is an arbitrary function in the exact solutions, which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.展开更多
In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensionai rotationai Euler equations for the incompressible fluid. The application of the method leads to a...In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensionai rotationai Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is＇of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh（ kt /2 ） and sech （kt/2） due to the rotational parameter k ≠ O. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth＇s rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.展开更多
基金Supported by NSFC for Young Scholars under Grant No.11101166Tianyuan Youth Foundation of Mathematics under Grant No.11126244+1 种基金Youth PhD Development Fund of CUFE 121 Talent Cultivation Project under Grant No.QBJZH201002Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No.KM201110772017
文摘In this paper, the separation transformation approach is extended to the （N ＋ 1）-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of SHe superfluid. This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation. Then the general solutions of the set of partial differential equations are obta/ned and the nonlinear ordinary differential equation is solved by F-expansion method. Finally, many new exact solutions of the （N ＋ 1）-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation. For the case of N 〉 2, there is an arbitrary function in the exact solutions, which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.
基金Supported by the National Natural Science Foundation of China under Grant No.11301269Jiangsu Provincial Natural Science Foundation of China under Grant No.BK20130665+2 种基金the Fundamental Research Funds KJ2013036 for the Central UniversitiesStudent Research Training under Grant No.1423A02 of Nanjing Agricultural Universitythe Research Grant RG21/2013-2014R from the Hong Kong Institute of Education
文摘In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensionai rotationai Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is＇of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh（ kt /2 ） and sech （kt/2） due to the rotational parameter k ≠ O. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth＇s rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.
